Physics 115 Lecture 9

Standing waves in an air

column

February 9, 2018

2

String fixed at both ends

Always a node at each end

Harmonics

Animated applet

Video

Chart of the harmonicsHarmonic #antinodes #nodes λ f

fundamental 1 2 f1

second 2 3 2 f1

third 3 4 3f1

2

1

L

2

2

L

2

3

L

stringc f

The recipe of harmonics The relative amplitudes (loudness) of the harmonics give

a musical instrument its characteristic tone quality

source: soundcheckucc.wordpress.com

Actual guitar note spectrum (Nonlinear scale. Click on

computermusicresource.com source page to hear the sound)

2

3

4 5 6 7 8 9

Wave reflection in an air column

A typical brass or woodwind instrument

creates a standing wave in an air column

that is closed at one end and open at the

other.

The standard diagrams can be confusing:

Animated longitudinal standing wave

Animation courtesy of Dr. Dan Russell, Grad. Prog. Acoustics, Penn State

Why does sound reflect from the open

end of a tube?

The formation of a

standing wave

requires counter

propagating waves,

which implies sound

reflection from the

open end of the

tube!

Source: hyperphysics

Acoustic impedance A sound wave reflects from the open end of an

air column because of the sudden change in

acoustic impedance Z (see link 5-09) (another tutorial)

Specific acoustic impedance (z) is the opposition

of a medium to wave propagation, and it depends

on the properties of the medium.

The acoustic impedance (Z) specifies the

impedance offered by an enclosed volume of a

medium of cross-sectional area A.

z cZ

A A

Acoustic impedance

ρ = density of medium (kg/m3). A denser medium

offers more resistance to wave propagation.

c = speed of sound (m/s). Faster sound speed

corresponds to a stiffer medium and more

resistance to wave propagation.

A = cross-sectional area (m2). A larger cross-

sectional area offers less resistance to wave

propagation. The impedance change at the end

of a tube is a result of the sudden change in the

cross-sectional area, and a sound wave will

reflect off this boundary.

z cZ

A A

Acoustic impedance z c

ZA A

Flaring the tube end

makes the impedance

change more gradual,

resulting in less

internal reflection and

a louder emitted sound

A sound wave travels through a cast iron (c = 3780 m/s,

ρ = 7700 kg/m3) rod and into a steel (c = 5100 m/s,

ρ = 7700 kg/m3) rod of the same diameter. As it does

so, it encounters _____ acoustic impedance.

A. a higher

B. a lower

C. no change in the

A sound wave travels through a cast iron (c = 3780 m/s,

ρ = 7700 kg/m3) rod and into a steel (c = 5100 m/s,

ρ = 7700 kg/m3) rod of the same diameter. As it does

so, it encounters _____ acoustic impedance.

A. a higher

B. a lower

C. no change in the

z cZ

A A

No change in ρ or A. The higher

speed of sound in steel means it is a

stiffer medium that offers more

resistance to wave propagation.